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Difference between revisions of "Irregular singular point"

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A concept that arose in the analytic theory of linear ordinary differential equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526801.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526802.png" />-matrix that is holomorphic in a punctured neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526803.png" /> and that has a singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526804.png" />.
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A concept that arose in the analytic theory of linear ordinary differential equations. Let $A(t)$ be an $(n\times n)$-matrix that is holomorphic in a punctured neighbourhood of $t_0\neq\infty$ and that has a singularity at $t_0$.
  
The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526805.png" /> is then called a singular point of the system
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The point $t_0$ is then called a singular point of the system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\dot x=A(t)x.\tag{*}$$
  
There are two non-equivalent definitions of an irregular singular point. According to the first one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526807.png" /> is called an irregular singular point of (*) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526808.png" /> has a pole of order greater than one at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i0526809.png" /> (cf. [[Analytic theory of differential equations|Analytic theory of differential equations]], as well as [[#References|[2]]]). According to the second definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268010.png" /> is called an irregular singular point of (*) if there is no number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268011.png" /> such that every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268012.png" /> grows not faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268013.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268014.png" /> along rays (cf. [[#References|[3]]]). The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268015.png" /> can be reduced to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268016.png" /> by the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052680/i05268017.png" />. An irregular singular point is sometimes called a strongly-singular point (cf., e.g., [[Bessel equation|Bessel equation]]). In a neighbourhood of an irregular singular point the solutions admit asymptotic expansions; these were studied by H. Poincaré for the first time [[#References|[1]]].
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There are two non-equivalent definitions of an irregular singular point. According to the first one, $t_0$ is called an irregular singular point of \ref{*} if $A(t)$ has a pole of order greater than one at $t_0$ (cf. [[Analytic theory of differential equations|Analytic theory of differential equations]], as well as [[#References|[2]]]). According to the second definition, $t_0$ is called an irregular singular point of \ref{*} if there is no number $\sigma>0$ such that every solution $x(t)$ grows not faster than $|t-t_0|^{-\sigma}$ as $t\to t_0$ along rays (cf. [[#References|[3]]]). The case $t_0=\infty$ can be reduced to the case $t_0=0$ by the transformation $t\to t^{-1}$. An irregular singular point is sometimes called a strongly-singular point (cf., e.g., [[Bessel equation|Bessel equation]]). In a neighbourhood of an irregular singular point the solutions admit asymptotic expansions; these were studied by H. Poincaré for the first time [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur les intégrales irrégulières des équations linéaires"  ''Acta Math.'' , '''8'''  (1886)  pp. 295–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Wasov,  "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur les intégrales irrégulières des équations linéaires"  ''Acta Math.'' , '''8'''  (1886)  pp. 295–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Wasov,  "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.A. Coddington,  N. Levinson,  "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR></table>

Revision as of 18:24, 21 August 2014

A concept that arose in the analytic theory of linear ordinary differential equations. Let $A(t)$ be an $(n\times n)$-matrix that is holomorphic in a punctured neighbourhood of $t_0\neq\infty$ and that has a singularity at $t_0$.

The point $t_0$ is then called a singular point of the system

$$\dot x=A(t)x.\tag{*}$$

There are two non-equivalent definitions of an irregular singular point. According to the first one, $t_0$ is called an irregular singular point of \ref{*} if $A(t)$ has a pole of order greater than one at $t_0$ (cf. Analytic theory of differential equations, as well as [2]). According to the second definition, $t_0$ is called an irregular singular point of \ref{*} if there is no number $\sigma>0$ such that every solution $x(t)$ grows not faster than $|t-t_0|^{-\sigma}$ as $t\to t_0$ along rays (cf. [3]). The case $t_0=\infty$ can be reduced to the case $t_0=0$ by the transformation $t\to t^{-1}$. An irregular singular point is sometimes called a strongly-singular point (cf., e.g., Bessel equation). In a neighbourhood of an irregular singular point the solutions admit asymptotic expansions; these were studied by H. Poincaré for the first time [1].

References

[1] H. Poincaré, "Sur les intégrales irrégulières des équations linéaires" Acta Math. , 8 (1886) pp. 295–344
[2] W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[3] E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
How to Cite This Entry:
Irregular singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irregular_singular_point&oldid=33054
This article was adapted from an original article by Yu.S. Il'yashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article